The up/down movement of the suspension will cause the crank axles to move back and forth relative to the seat (just because the geometry of the ‘frame’ is changing with the extension/compression of the suspension).
In addition, I think the up and down movement will also translate to rotation of the cranks (think of how the sprocket on the hub moves in relation to the centre of the crank axle when there is linear movement up and down on the suspension) – so the cranks are still tied to the up-down movement of the suspension and in essence it is not any different to if they were connected directly to the hub axle (which is the degenerate case of your geometry).
If you want something that would work to dissociate the cranks from the suspended hub axle, think of how you could efficiently make a vertical splined shaft drive with the splined prop-shaft moving parallel to the suspension movement (say with a ball spline). You’d have to efficiently have two right angle drives to change rotation to vertical and back, and do this on both sides, probably via a jack-start arrangement…..